The Catholic parish records for Dromtarriffe start in 1832, both for baptisms and marriages, and are available online at this link: https://registers.nli.ie/parishes/0047
I looked in the 1832-1835 time period for Jeremiah's baptism, but did not find it. That doesn't mean that it isn't there, however, for two reasons: (i) some entries are faded and difficult to read, and (ii) in those days people often did not know exactly when they were born, so ages given in overseas records after they emigrated may just be estimates. You can download pages from the register at that site, and if you decide to look for him in 1836 or later, note that the entries follow this pattern (in that parish at that time):
In the left column, it will say something like: "Jeremia[h] [Denahy]", then to the right it will say: "[Idem] Bapt. Jeremiam Denahy F. [filium] Phillipi Denahy & Sarah O'Leary" ("[On the same day] I baptized Jeremiah Denahy, son of Phillip Denahy and Sarah O'Leary"). The parts which I've shown in brackets may or may not be there. By the way, the "Phillipum" (objective) form of Phillip's name which you found would be used in his own baptismal record, just as "Jeremiam" would be used in the record for Jeremiah.
The mother's name will be followed by the name of the townland where the family lived, and the names of the two sponsors at the ceremony will be given in the right-hand column. As you may have learned already, some priests chose to create Latinate versions of given names, or to use Biblical names which seemed close. Thankfully, they did not attempt to alter the surnames, although different priests spelled them differently at times, because they didn't know (or care) how to write them in Irish. The Irish form of the surname Denahy/Dennehy is Ó Duineacha. It's a name which originated in Cork and is still found mostly there. Since that area would have been overwhelmingly Irish-speaking at the time, it is likely that Jeremiah was actually called Diarmuid (Dermot) by the family, and his parents may have been called Feidhlim (Phelim) and Sorcha (or Sadhbh). Some of those Latinate/Biblical names stuck eventually, and some didn't. The use of Dionysius for Donncha never took, but Denis as an alternative did make it into general use.
I was able to find the marriage record for Jeremiah Denahy and Ellen Driscoll, however (copy attached). They were actually married on July 10, 1852, not June 10. Whoever created the transcript you found wasn't paying close attention to the names of the months. The entry for them is the second one on the left-hand side of the attached image. It also lists the townland of Cloonbannin. That is probably where the Denahy's lived (i.e., where the couple were going to live), but I’m not certain. The townland is often not stated in marriage records, and when it is the townland for each person is sometimes given, so perhaps they were both from the same townland. The record also shows the witnesses at the marriage: John [Joannes] Denahy and Conor [Cor's = Cornelius] Driscoll, who would probably have been siblings or cousins of the groom and bride, respectively.
There are actually two townlands called Cloonbannin in the Catholic parish of Dromtarriffe: Cloonbannin West (in Irish, Cluain Báinín Thiar) and Cloonbannin East (in Irish, Cluain Báinín Thoir), which are located next to one another. It isn't clear from the marriage record which one was their home, and there may actually have been only one Cloonbannin at the time (names and boundaries sometimes changed). At this link, you can see a map and info for Cloonbannin West, with a link there to see the same info for the other townland: https://www.townlands.ie/cork/duhallow/cullen/skagh/cloonbannin-west/
Both townlands are located in the civil parish of Cullen. While the Catholic parish is called Dromtarriffe, it contains several civil parishes, which were essentially administrative districts by then. Within each civil parish there are a number of townlands, sometimes just a dozen or so, but sometimes many more. These are the smallest official districts, and are still used as addresses in many cases, since people in rural areas usually lack street addresses. The spelling of townland names sometimes changed over the years, so records for the same family may have different forms of the townland name. One common reason for variations in the Anglicized spelling of parish and townland names is that the officials writing them down often did not understand Irish, and in some cases had difficulty hearing correctly what was being said when the names were spoken by Irish speakers (the Irish language has a number of sounds which don’t exist in English, and are often misheard), or they based the anglicized form on the Irish spelling, without understanding how to pronounce the Irish letter combinations.
I was particularly interested in your query, because one of my great-grandmothers grew up in the Catholic parish of Dromtarriffe, in a different townland, but not far from Cloonbannin. I also noticed from my own genealogical notes that, in the 1841 baptismal record for one of my distant cousins, the sponsors were John Denehy and Catherine Denehy (maybe the same John Denehy mentioned above?), who would likely have been relatives or close friends. If you've done dna testing and want to compare results, let me know. Also, in case you're interested, the name Dromtarriff[e] comes from the Irish Drom Tarbh, meaning "bull ridge", and may refer to an ancient pagan ceremony where a bull calf was sacrificed on May 6th near a holy well still located in the parish.
